# De Moivreâ€™s Theorem

- If
*n*is any rational number, then (cos*Î¸*+*i*sin*Î¸*)^{n}^{ }= cos*nÎ¸*=*i*sin*nÎ¸**.* - If
*z*= (cos*Î¸*_{1}+*i*sin*Î¸*_{1}) (cos*Î¸*_{2}+*i*sin*Î¸*_{2}) (cos*Î¸*_{3}+*i*sin*Î¸*_{3}) â€¦ (cos*Î¸*+_{n}*i*sin*Î¸*) then_{n}*z*= cos (*Î¸*_{1}+*Î¸*_{2}+*Î¸*_{3}+ â€¦ +*Î¸*) +_{n}*i*sin (*Î¸*_{1}+*Î¸*_{2}+*Î¸*_{3}+ â€¦ +*Î¸*), where_{n}*Î¸*_{1},*Î¸*_{2},*Î¸*_{3}â€¦*Î¸*âˆˆ_{n}*R.* - If
*z*=*r*(cos*Î¸*+*i*sin*Î¸*) and*n*is*z*^{1/n}=*r*^{1/n}*k*= 0, 1, 2, 3, â€¦, (*n*â€“ 1).If*Deductions:**n*âˆˆ*Q*, then- (cos
*Î¸*â€“*i*sin*Î¸*)= cos^{n}*nÎ¸*â€“*i*sin*nÎ¸* - (cos
*Î¸*+*i*sin*Î¸*)^{â€“n}= cos*nÎ¸*â€“*i*sin*nÎ¸* - (cos
*Î¸*â€“*i*sin*Î¸*)^{â€“n}= cos*nÎ¸*+*i*sin*nÎ¸* - (sin
*Î¸*+*i*cos*Î¸*)^{n}*n*

- (cos

This theorem is not valid when

*n*is not*a**rational number or the complex number is not in the form of cos**Î¸*+*i*sin*Î¸**.*# The nth root of unity

Let

*x*be the*n*th root of unity. Then*x*= 1

^{n}= 1 +

*i*(0)= cos 0Â° +

*i*sin 0Â°= cos (2

*k**Ï€*+ 0) +*i*sin (2*k**Ï€*+ 0)= cos 2

*k**Ï€*+*i*sin 2*k**Ï€*(where*k*is an integer)â‡’

*x*= cos*k*= 0, 1, 2, â€¦,*n*â€“ 1Let

*Î±*= cos (2*Ï€*/*n*) +*i*sin (2*Ï€*/*n*). Then the*n*th roots of unity are*Î±*(^{t}*t*= 0, 1, 2, â€¦,*n*â€“ 1), i.e., the*n*th roots of unity are 1,*Î±*,*Î±*^{2}, â€¦,*Î±*^{n}^{ â€“ 1}.